Entire Functions

282 Pages·1954·6.415 MB·English

by  Ralph Philip Boas

Entire Functions
RALPH PHILIP BOAS, JR.
Northwestern University, Euanston, IZZinois
I9S4
ACADEMIC PRESS INC., PUBLISHERS
NEW YORK, N. Y.
Copyright, 1954, by
ACADEMIC PRESS INC.
125 East 23rd Street, New York 10, N. Y.
All Rights Reserved
NO PART OF THIS BOOK MAY BE REPRO-
DUCED IN ANY FORM, BY PHOTOSTAT,
MICROFILM, OR ANY OTHER MEANS,
WITHOUT WRITTEN PERMISSION FROM
THE PUBLISHERS.
Library of Congress Catalog Card Nuvnber: 6441061
PRINTED IN THE UNITED STATES OF AMERICA
TO MARY
PREFACE
My chief aim has been to give an account of the extensive modern
theory of functions of exponential type; the natural domain for these
functions is often a half plane or an angle rather than the whole plane.
Thus this book is not a comprehensive treatise on entire functions, and is
not concerned exclusively with entire functions. However, a short and
reasonably accurate title seemed preferable to a longer and more descrip-
tive one. Even the limited subject of entire functions is too vast to be
dealt with in a single volume with any approach to completeness, and
I have preferred to omit altogether those topics which I could not cover
fairly thoroughly. Thus there is no mention at all even of such important
matters as Picard’s theorem and the whole circle of ideas connected with
it. Functions of exponential type have many applications in other fields;
to cover the applications adequately would demand a book in itself, but
I have discussed selected examples from a variety of fields to indicate how
some of the applications arise.
I begin, in Chapter 2, with an account of some of the fundamental but
elementary results on entire functions of finite order in general; much of
this material is to be found in almost any comprehensive book on the the-
ory of functions. Otherwise I have selected mostly material which is not
covered in detail in existing books.
A reader familiar with the theory may notice a number of simplifications
of proofs and slight improvements of theorems, as well as an attempt to
use unified methods wherever possible. However, I do not claim for my
own all material which is not explicitly credited to somebody else, since
I have not attempted to locate the original sources for many things which
are well known to workers in the field.
I assume that the reader is familiar with the basic theory of functions
of a complex variable, as presented in any modern text intended for mathe-
maticians; I assume that he knows, or is willing to look up, such things as
Jensen’s and Carleman’s theorems, the ideas associated with the names of
Phragmh and Lindelof (although I state the theorems for reference), and
the elements of the theory of harmonic and subharmonic functions in the
plane. I also assume a certain command of the tevhnique of “hard” anal-
ysis: I take for granted manipulations with lim sup and lim inf, with 0
and 0, with Holder’s, Minkowski’s arid Jensen’s inequalities, and with
Lebesgue and Stieltjes integrals. There no longer aeems to be any justi-
fication for depriving one’s self of the ronvenience of the Lebesgue integral
even though almost everything in the theory of entire functions can be done
without it. At a few points I require the L2 theory of Fourier transforms.
I have not tried to quote all papers connected with a given topic nor
have I tried systematically to assign theorems to their discoverers. Names
like “P6lya’s theorem” are intended merely as vatch phrases to identify
results that are frequently referred to. The bibliography contains only
those works which are referred to in the text. Bibliographies up to the
middle 1920’s are given by Valiron [3], [5],a nd for older work I usually
refer to Valiron rather than to the primary sources. Would-be investigators
in the field of entire functions are warned that the non-inclusion of a given
topic or specific problem in this book, even when it may seem germane to
topics which are included, is no guarantee that it is not already discussed
in the literature. The field has already suffered more than most from
repeated rediscovery of results and apparent rehic.tance of investigators
to read each other’s writings.
The reader will find only one or two new pieceh of terminology, arid no
abbreviations (other than standard symbols). The temptation to introduce
ad hoc abbreviations and portmanteau words is almost overwhelming
when one works for a long time on a subject, but I believe that the saving
of pencil for the author and of type for the printer is far offset by the in-
convenience for the reader, especially in a book whirh is intended to be
ronsulted by the non-specialist in search of possibly applicable results as
well as to be studied systematically by a student who desires to berome
acquainted with the field.
Theorems which are stated but not proved in the book are identified
by stars; some of these are suitable for exercises while others are too wm-
plicated to have their proofs included.
The reference numbers attached to forinulas, theorems, etc., are intended
to be read as integers in the scale of 100, with the dots indicating the spave
between “digits.” Superscript letters refer to the notes at the ends of the
chapters.
I am indebted to the John Simon Guggenheim Memorial Foundation
and to Northwestern University for financial support during the aca-
demic year 1951-52, when most of the book was written.
K. P. BOAS,J R.
Bvariston, Illinois
June, 1954
CHAPTER1
INTRODUCTION
1.1. Terminology. In this chapter I shall define some terms and intro-
duce some notations which will be used consistently; and collect, in most
cases without proof, some auxiliary theorems which will be used more
or less frequently but do not altogether belong to the subject matte1 of
the book.
A function of the complex variable z will be called regular in a region if
it is analytic and single-valued there. An entire function” is one which is
+
regular for all finite z. We consistently write z = x iy = reie, and under-
stand that x, y, r, 0 have these meanings unless something is said to the
+
contrary; similarly, writing z1 implies that z1 = x1 iy,, etc. We write
I = z - iy, but f(z) means the function such that 3(2) = f(x). We call T
the modulus and 0 = arg z the argument of z. We also write x = Nz),
y = 3(z). “The upper half plane” means the half plane y > 0 or y 2 0
>
according to context; similarly “the right-hand half plane” is 2 0 or
z 2 0, and “the unit circle” is I z I < 1 or I z I 5 1. A contour is a recti-
fiable Jordan arc or curve.
If f(z) is regular in I z I < R (or entire), M(T)d enotes the maximum
modulus of f(z) for I z I = T < R (or < a),a nd m(r) denotes the minimum
modulus for I z I = r. We write Mf(r),e tc., when it is necessary to call
attention to the particular function that is being considered. By n(r), or
nf(r),e tc., w1e d‘ enote the number of zeros of f(z) in I z I 5 r, and by N(r)
the integral t-’n(t) dt, provided that n(0) = 0. The zeros themselves are
frequently denoted by zk = rke i9b and are supposed to be arranged in order
of increasing modulus, with multiple zeros counted according to their
multiplicities.
A number appearing without explanation in a formula is defined by the
formula. We occasionally use the same symbol A to mean different num-
bers in different parts of the same formula or chain of formulas; any am-
biguity can be avoided, if desired, by the reader’s numbering such A’s
serially as they occur. Greatest lower bounds and least upper bounds are
denoted by inf and sup, respectively. When t > 0, log’t means sup (log t, 0)
[XI
and log-t means inf (log t, 0); similarly for other functions. By we mean
the greatest integer not exceeding x, except when it is clear from the con-
-
text that the brackets are used merely as parentheses.
The statements f(x) = O{g(z)), f(x) = o{g(x)),f (x) g(x) mean, as
usual, that f(z)/g(x) is bounded, approaches zero, or approaches 1, re-
1
2 ENTIRE FUNCTIONS
spectively; and f(x) 5 O(g(x)),f (x) 5 o(g(x))m ean that f(x)/g(x) is
bounded above by a constant or by o(1).
-00 I"
The notation f(x)d x means limR+, f{(xr)d -x. 'T he principal value
[ Ib}
+
(at c) of f(z) dx, a < c < b, means lim,,o f(x)d x. The
class Lp consists of the measurable functions f(x) for wc+ht ich I f(x) I* is
integrable (over whatever set is under consideration).
1.2. Jensen's, Carleman's and Nevanlinna's formulas. These are for-
mulas connecting the zeros of f(z) with its behavior on the boundary of
a circle or a half plane.
1.2.1. Jensen's Theorem.' If f(z) is regular in r < R, and f(0) # 0, then
for r < R we have
1
2*
N(r) = (2~)-' log I f(re") I d0 - log I f(0) I.
0
1.2.2. Carleman's Thcorem? If f(z) is regular for y 2 0 and f(0) = 1,
and if zk arc the zpros of f(z) in the upper half plane,
1.2.3. Nevanlinna's Theorem (Poisson's Formula for a Semicircle) If
.a
f(z) is regular for y 2 0 and zk are its zeros,
where
PI = t2 - 21tx -l- r2 - R4 - 2tRR22 x + r2t2'
We may obtain 1.2.2 formally from 1.2.3 by letting z 0.
---f
1.3. Carathbodory's inequality.&T his gives an upper bound for the
modulus of an analytic function in a circle when we know an upper bound
for its real part (not the absolute value of its real part) in a larger circle.
1.3.1. Theorem. If f(z) is regular in I z 1 5 R and %{f(z)) 5 Q(r) in
INTRODUCTION 3
I z I 2 r thenfor 0 < r < R
We shall need the result of applying this to the logarithm of a zero-free
function.
1.3.2. Theorem. Let f(z) be regular in I z I 5 R and have no zeros there. If
f(0) = 1,
(1.3.3) log m(r) 2 - O<r<R.
CarathBodory 's inequality gives
taking the z of modulus r for which I f(z) I = m(r),w e have (1.3.3).
CarathBodory's inequality also furnishes a convenient means of dealing
with the analogue of Liouville's theorem in which the real part of the func-
tion is considered instead of the modulus.
1.3.4. Theorem. Iff(z) is entire and %(f(z)} 5 A(e)rP" on arbitrarily
large circles for each positive e, f(z) is a polynomial of degree at most p.
Supposing for convenience that f(0) = 0, we have by 1.3.1, for r = R/2,
where R is one of the values of r for which the hypothesis is satisfied,
If(z) I Izp +l+' ~ ( erP) + ',
and so by Liouville's theorem f(z) reduces to a polynomial of degree at mos t p.
1.4. Phragmh-Lindelof theorems.& These are generalizations of the
maximum principle in which we infer the boundedness of a function inside
an unbounded region from the hypotheses that the function is bounded
on the boundary and not of too rapid growth inside.
1.4.1. Theorem. Let f(z) be regular in the half plane x > 0, continuous in
the closed half plane, I f(iy) 1 IM and f(z) = O(ers),p < 1, uniformly in
8, for a sequence r = rn -+ 00. Then I f(z) I 5 M for x 2 0.
Consider F(z) = f(z) exp (-ezy), where p < y < 1 and E > 0. Then
I F(z) I 5 I f(z) I exp (-cry cos ye). Since y < 1, cos ye > 0 in the half
plane and so I F(iy) I 5 I f(iy) I 5 M. On the arc I 8 I 5 T of I z I = r,, ,
I F(z) I 5 I f(z) I exp (- ernYc os 7ry/2), and the right-hand side approaches
zero as r, -+ 00 because y > p. Hence if r, is large enough, I F(z) 1 5 M
on 1 z I = r, , I 8 I IW T, and so I F(z) I IM in the semicircular region
which this arc cuts from the half plane, and hence in the entire half plane
since rn can be arbitrarily large. That is, I f(z) I IM exp (cry)a nd we may
let r -+ 0 (for each fixed z) to obtain the conclusion of 1.4.1.
4 ENTIRE FUNCTIONS
Of course the theorem is true for any half plane, since an arbitrary half
plane may be translated and rotated to become x > 0. In addition, the
theorem may be transformed by conformal mapping into one for any
angle. We give the general statement since we shall need to use it.
1.4.2. Theorem. Let f (2) be regular in the angle D : I arg z I < $5 ~/ac,o n-
I <
tinuous in the closed angle, If(z) M on the boundary, f(z) = O(er8),
0 < a, uniformly in 0, for r = r,, --+ m. Then I f(z) I IM throughout D.
The conclusion of 1.4.1 still holds under a more general hypothesis.
1.4.3. Theorem. Letf(z) be regular in z > 0, continuous in z 2 0, I f(iy) I <
M and f(z) = O(ear)r, -+ w ,for z = reaa,w here a is a number between -r/2
and 7r/2. Then I f(z) I < Me(oaeac) rcos ' for x 2 0. In particular, I f(z) I 5 M
if the hypotheses are satisfied with arbitrarily small positive a;a nd f(z) = 0
if they are satisfied with arbitrarily small a, i.e. if f(reia) = O(e-'u(r)),
o(r)
3 00.
Now we consider F(z) = e-braeoaf (z), b > a, which is bounded by M on
the positive imaginary axis and by some number N on the positive real
faixrsist q(suinacder aFn(t,x a) n.--d) s0i maisl azrl y-+ I Fm()z.) BI <y 1m.4a.x2(,M I F, (Nz)) I i nI t hmea fxo u(rMth, qN u)a idnr atnhte.
Since I F(z) 1 = N at some finite point we must have N 5 M, since other-
wise \ F(z) I would take its maximum value in the right-hand half plane at
an interior point and so would be a constant, which could only be M. Since
now I f(x)e-bzEma I 5 M for all b > a, we can let b --+ a to obtain the
result.
1.4.4. Theorem. If f(z) -+ a as z -+ 00 along two rays, and f(z) is regular
and bounded in the angle between them, f (2) --+ a uniformly in the whole angle.
is WI aer gm xa yI sIupC#pI o<se Wwi tRh. oLuett loFs(sz o)f =ge nze(xr a+lit yX )t-'hf(za),t a =X >0 a 0n. dL tehta te t>he 0a nagnlde
take r1 so large that I f(re*") I < e for r 2 rl.T hen, with rl fixed, take X
so large that I F(z) I < e for r 5 r1 and I arg x I I9. B y 1.4.2, I F(z) I < e
throughout I arg x I I9 , and
+
I f(z) I I (1 X/r) I F(z) I < 2% r > A.
This establishes the result.
By conformal mapping we can obtain a similar result for a strip.
1.4.5. Corollary. If f(x) -+ a as x --+ 00 for y = y1 and y = y2 , and f(x) is
regular and bounded for x 2 b, y1 5 y 5 y2 , then f(z) -+ a uniformly for
y1 I y IY 2.
1.4.6. Beurling's Theorem. If f(x) is regular in x > 0 and continuous in
x 2 0, if1 f(iy) I I $J(/y I ), where +(r) is continuoirs and
lim sup rWPlo g 4(r) = 0, O < p < l ,
and I f(z) 1 = O(e") for a sequence rn -+ 00, for each e > 0, then either f(z)
INTRODUCTION 5
is bounded for x 2. 0 or
(1.4.7) log M(r) < log d(r).sec % ap
for a sequence r = R, -+ 00.
If 4(r) were bounded we should have f(z) bounded, by 1.4.3; hence we
may assume $(r) unbounded. Consider F(z) = f(z) exp (- exP - a),w here
zp is the branch which is positive for positive z. For z = iy we have
log I F(z) I 5 log 4( I y I ) - E cos % ~p - a.
Now take a = a(€)s o that
+
log 4(r) 5 rpc cos % ap a(€)
for all r, with equality for some r = r(c).T hen we have log I F(x) I 5 0 for
z = iy, and hence 1 F(z) 1 5 1 for x 2 0 by 1.4.3. Then for r = r(e) we
+ +
have log M(r) L erP a(€),l og 4(r) = E cos 14 7rp.r' a(€),a nd so
+
log 4(r) 2. cos % ~p.logM (r) a(€)( 1 - cos % ap).
Since r(E) --+ 00 as E + 0 (because 4(r) is unbounded), the conclusion fol-
lows.
A result very similar to 1.4.4 is
1.4.8. Montel's Theorem.b Zf f(x) is regular and bounded in the angle
between two rays, and f(z) -+ a as z -+ 00 on one ray in the interior of the
angle, then f(x) a uniformly in any interior angle.
-+
The equivalent result for a strip is
1.4.9. Theorem.b If f(z) is regular and bounded for x 2 b, y1 I y 5 y2,
and f(z) -+ a as x --+ co for y = y3 , y1 < y3 < y2 , then f(z) + a uniformly
for YI I Y I ~ 2 .
1.6. Density of sequences and sets. If {A,] is a nondecreasing sequence
of positive numbers it is said to have density D if limn-+mn/ A, = D (Dm ay
be 0, finite, or infinite). If the numbers X , may have either sign, {A,] has
density D if { I A, I } has density 20. We shall have frequent use for the
following elementary lemma.
1.5.1. Lemma. If {A,}: is a nondecreasing sequence of positive numbers and
n(r) denotes the number of A, not exceeding r, the statements n/A, D and
--+
n(r)/r -+ D are equivalent.
On one hand, n(A,)/A, = n/X, , so if n(r)/r -+ D, n/L -+ I). On the
other hand, if n/L -+ I), suppose that X, is the first X greater than A, ;
then A,-1 = A,, , so (m - l)/X, --+1 1. For X , < x < A,, n(x) = m - I,
so n(z)/x < m/X, I). Thus lim sup n(x)/x I D. Similarly
-+
lim inf n(x)/x 2 L).
Even if the sequence {A,), A,, > 0, fails to have a density, it always
has (finite or infinite) upper and lower densities, lim sup n/l, , lim inf n/h,, .